1. Consistent Mesh Parameterizations Peter Schröder Caltech Wim Sweldens Bell Labs Emil Praun Princeton

2. Motivation Digital Geometry Processing (DGP) Do for surfaces what DSP does for sound, images, and video Requires smooth parameterizations denoisingenhancement

3. Parameterizations Smooth sampling pattern Individual surface setting –coarse mesh (base domain) –semi-regular refinement Efficient algorithms progressive transmissionhierarchical editing irregularsemi-regular

4. Parameterizations What about multiple objects? Computing the mean... and many other algorithms –blending, principal components, etc. Need consistent parameterizations!... + ( _ 1 n ++ ) =

5. Goal Consistent parameterization same base domain correspondences –vertices, edges smooth & fair Common sampling samples 1-1 Base Domain Input Meshes with Features Semi- Regular Meshes DGP Applications

6. Previous Work Mesh Simplification, Progressive Meshes, … [Hoppe 94-98] MAPS, Morphing [Lee 98, 99] Disp. Subdivision Surfaces / Normal Meshes [Lee 2000] / [Guskov 2000]

7. Approach Identify feature points (user) Parameterize interior of patches Trace curves for base domain edges

8. Tracing Curves Net topologically equivalent to base domain Curves intersect only at vertices Same neighbor ordering around vertices

9. Tracing Curves Restricted “brush fire” (BFS traversal): Do not cross other curves Start & end in correct sector

10. Problem: Encircling To avoid, first trace spanning tree Proof of correctness in the paper a b c d e a b d c e Base domainMesh

11. Topological Equivalence Guarantee topological equivalence of traced net and base domain Trace curves w/ restricted brush fire Complete spanning tree before adding cycles

12. Is “Topological” Enough?

13. Swirl Operator Simple relaxation doesn’t undo swirls Infinity of possible configurations We want the least unnecessary swirls Optimization very hard; use heuristics

14. Heuristics 1.Feature points repel curves 2.Introduce curves in order of length 3.Delay edges of flipped triangles

15. 1.Features Repel Curves Use embedding in  n Compute I i (k) = “influence” of feature i on vertex k Contour plot

16. 1.Features Repel Curves Initialize: I i (i) = 1 I i (feature j) = 0 Relax for mesh surface Linear system Floater’s weights Contour plot

17. 1.Features Repel Curves Trace curve (a,b): brush fire with variable propagation speed 1 0 a b c k(0.2,0.1,0.4,…) a(1,0,0,0,..) b(0,1,0,0,..) c(0,0,1,..) k Priority P(k) = 1- I a (k) - I b (k)

18. 1.Features Repel Curves

19. 2.Prioritize Curves by Length First stage: complete spanning tree Second stage: complete whole net For each stage, keep priority queues Queues contain candidate curves May need to update to enforce topology

20. 3.Delay Edges of Flipped Triangles a b c a cb a bc OR OK Flipped  “swirl detector”

21. 3.Delay Edges of Flipped Triangles “swirl detector” ?ab c Flippedac b a c b a c b ac b

22. Edge Straightening Mesh Temporary 2D parameterization

23. Edge Straightening Mesh Temporary 2D parameterization

24. Examples

26. Principal Mesh Components

27. Texture Transfer

29. Detail Transfer

30. N-way Shape Blending Horse.5 Man.25 Cow.25 Horse.25 Man.25 Cow.5 Horse.25 Man.5 Cow.25 Horse.33 Man.33 Cow.33

31. Future Work Higher genus, boundaries, missing feature points, additional feature points. Transfer of animation controls Use of principal component analysis for indexing and recognition in large database Compression of multiple shapes

32. Acknowledgements Support: Bell Labs Models: Cyberware, Stanford, Freiburg U. Code: Igor Guskov Help: Adam Finkelstein, Tom Funkhouser, Lee Markosian, & the Princeton crowd

33. Proof Spanning tree prevents encircling Before tree completion, no cycles Base domain Mesh a a b b

34. Proof Spanning tree prevents encircling Before tree completion, no cycles Base domain Mesh a a b b